Regular subdivision Loop 87 Catmull&Clark 78 Zorin&Schröder 01 Use split&tweak curve subdivision to create smooth surfaces Presti

T-mesh subdivision Assume Triangle mesh with arbitrary genus

Lecture Objectives Learn how to implement a regular subdivision for manifold triangle meshes using a butterfly scheme to compute the new vertices at each stage. Reading: Lear how to extend the above to an adaptive subdivision by studying the research paper: “A Sub-Atomic Subdivision Approach” by S. Seeger, K. Hormann G. Hausler and G. Greiner

Subdivision Insert a new vertex in the middle of each edge Split each triangle into 4 triangles

An example of 2 iterations on a tetrahedron Repeat the operation several times BUT, at each step, you will adjust the position of the new vertices, using the butterfly rule

Butterfly mask To compute the position of the new (mid-edge) point, multiply its neighbors by the corresponding coefficients and add them up.

A more intuitive formulation First insert the new points in the middle of the edges m = average(f) Then compute adjustment vectors d = (average(s) – average(t))/4 Then adjust all the m points by displacing them by the corresponding d m = m + d Then compute the new triangles m f s t equivalent

Subdivision a triangle mesh k times Read the input file and construct the X, Y, Z, V tables Allocate X[N4k], Y[N4k], and Z[N4k] and fill the first N entries Repeat the following process k times Compute the O table For each corner c, find the opposite corner b, then O[c]:=b; O[b]:=c; Compute mid-edge points For each corner c do n+=1; (X[n],Y[n],Z[n]):=average(f)+(average(s) – average(t))/4; M[c]:=M[c.o]:=n; Compute a new V-table (4 times longer than the previous one) V[n++]=c.v; V[n++]=c.p.m.v; … // 12 entries Write the X, Y, Z and the V tables on file

Input/output (file) format N # the number of vertices x[0] y[0] z[0] x[1] y[1] z[1] … x[N-1] y[N-1] z[N-1] T # the number of triangles V[0] V[1] V[2] # the Integer IDs of vertices for triangle 0 V[3] V[4] V[5] # the Integer IDs of vertices for triangle 1 V[3] V[4] V[5] # the Integer IDs of vertices for triangle 2 V[6] V[7] V[8] V[3T-3] V[3T-2] V[3T-1] # the Integer IDs of vertices for triangle T-1

How to construct the O table FOR c:=0 TO 3T-2 DO FOR b:=c+1 TO 3T-1 DO IF THEN { O[c]:=b; O[b]:=c} (c.n.v==b.p.v) && (c.p.v==b.n.v)

How to access the neighbors from a corner Given a corner c, write down the complete expression for m(c) = average(f) + (average(s) – average(t))/4 using .n, .p, .o, .l, .r, and .v operators c.o.l.v c.v m(c) c.p.v c.n.v c.o.v c.r.v c.l.v c.o.r.v c s m(c) f t c m(c)= (c.n.v+c.p.v)/2+((c.v+c.0.v)/2–(c.l.v+c.r.v+c.o.l.v+c.o.r.v)/4)/4

How to split a triangle Given a corner c, write down the complete expression for the next 12 entries of V as shown using .n, .p, .o, .l, .r, and .v operators c.m.v c.v c.p.m.v c.n.m.v c.n.v c.p.v 1 2 3 4 5 6 7 8 9 10 11 12 c

Generating the V entries for 4 new triangles Write the 12 V-table entries in terms of c c.v c.p.m.v c.n.m.v c.n.v c.m.v c.p.v c..n.m.v c.m.v c.v c.p.m.v c.n.m.v c.n.v c.p.v 1 2 3 4 5 6 7 8 9 10 11 12

Complete T-mesh subdivision algorithm read k; allocate(X[4k], Y[4k], Z[4k]); for i:=0 to k–1 do read(X[i],Y[i],Z[i]); read t, allocate(V[3t]); for i:=0 to 3t–1 do read(V[i]); Allocate(O[3t],F[3t],M[3t],W[12t]); for c:=0 to 3t–2 do for b:=c+1 to 3t-1 do if (c.n.v==b.p.v) && (c.p.v ==b.n.v) then {O[c]:=b; O[b]:=c}; i:=k–1; for c:=0 to 3t–1 do {M[c]:=M[c.o]:=++i; (X[i],Y[i],Z[i]):=(c.n.v+c.p.v)/2+((c.v+c.o.v)/2–(c.l.v+c.r.v+c.o.l.v+c.o.r.v)/4)/4}; i:=0; for c:=0 to 3t–1 do if c.f then { F[c]:=F[c.n]:=F[c.p]:=false; W[i++]:= c.v; W[i++]:= c.p.m.v; W[i++]:= c.n.m.v; W[i++]:= c.p.m.v; W[i++]:= c.n.v; W[i++]:= c.m.v; W[i++]:= c.n.m.v; W[i++]:= c.m.v; W[i++]:= c.p.v; W[i++]:= c.m.v; W[i++]:= c.n.m.v; W[i++]:= c.p.m.v}; write (4k, X, Y, Z, 4t, W);

Rebuilding the O table directly (Stephen Ingram) # this program takes a T-mesh and computes a subdivided T-mesh using the butterfly subdivision read k; allocate(X[4k], Y[4k], Z[4k]); for i:=0 to k-1 do read(X[i],Y[i],Z[i]); # k = number of vertices read t, allocate(V[3t], O[3t]); for i:=0 to 3t-1 do read(V[i]); # t = number of triangles for i:=0 to 3t-1 do read(O[i]); #read in opposites from file allocate(nO[12t], F[3t], M[3t], W[12t]); # F = true/false it has been fourthed, M = midpoints, nO is new Opposite array # compute each point's midpoint and its opposite point's midpoint (the same) i:=k-1; for c:=0 to 3t-1 do {M[c]:=M[c.o]:=++i; (X[i],Y[i],Z[i]):=(c.n.v+c.p.v)/2+((c.v+c.o.v)/2-(c.l.v+c.r.v+c.o.l.v+c.o.r.v)/4)/4}; # compute triangles for new surface i:=0; for c:=0 to 3t-1 do if c.f then { F[c]:=F[c.n]:=F[c.p]:=false; # to avoid recalculating a new triangle W[i]:= c.v; nO[i++]:=c.m.v; W[i]:= c.p.m.v; nO[i++]:=c.n.o.p.m.v; W[i]:= c.n.m.v; nO[i++]:=c.p.o.n.m.v;# W = array of new triangles W[i]:= c.p.m.v; nO[i++]:=c.p.v; W[i]:= c.n.v; nO[i++]:=c.n.m.v; W[i]:= c.m.v; nO[i++]:=c.v; W[i]:= c.n.m.v; nO[i++]:=c.n.v; W[i]:= c.m.v; nO[i++]:=c.n.o.n.m.v; W[i]:= c.p.v; nO[i++]:=c.p.m.v; W[i]:= c.m.v; nO[i++]:=c.v; W[i]:= c.p.m.v; nO[i++]:=c.p.v; }; write (4k, X, Y, Z, 4t, W, nO);

Subdivision in Processing: Corners int t (int c) {int r=int(c/3); return(r);}; int n (int c) {int r=3*int(c/3)+(c+1)%3; return(r);}; int p (int c) {int r=3*int(c/3)+(c+2)%3; return(r);}; int v (int c) {return(V[c]);}; int o (int c) {return(O[c]);}; int l (int c) {return(o(n(c)));}; int r (int c) {return(o(p(c)));}; // indices to mid-edge vertices associated with corners int w (int c) {return(W[c]);}; boolean border (int c) {return(O[c]==-1);}; // if has no opposite // shortcut to get the point of the vertex v(c) of corner c pt g (int c) {return(G[V[c]]);};

Subdivision in Processing: O table // sets O table from V, assumes consistent orientation of triangles void computeO() { // init O table to -1: has no opposite (i.e. is a border corner) for (int i=0; i<3*nt; i++) {O[i]=-1;}; // for each corner i, for each other corner j for (int i=0; i<3*nt; i++) { for (int j=i+1; j<3*nt; j++) { if( (v(n(i))==v(p(j))) && (v(p(i))==v(n(j))) ) { O[i]=j; O[j]=i;};};}; // make i and j opposite if they match };

Subdivision in Processing: Split edges // creates a new vertex for each edge and stores its ID // in the W of the corner (and of its opposite if any) void splitEdges() {for (int i=0; i<3*nt; i++) { // for each corner i if(border(i)) {G[nv]=midPt(g(n(i)),g(p(i))); W[i]=nv++;} else {if(i<o(i)) {G[nv]=midPt(g(n(i)),g(p(i))); W[o(i)]=nv; W[i]=nv++; }; }; // if this corner is the first to see the edge };

Subdivision in Processing: Bulge edges // tweaks new mid-edge vertices according to the Butterfly mask void bulge() { for (int i=0; i<3*nt; i++) { // no tweak for mid-vertices of border edges if((!border(i))&&(i<o(i))) { G[W[i]].addScaledVec(0.25, midPt(midPt(g(l(i)),g(r(i))), midPt(g(l(o(i))),g(r(o(i)))) ).vecTo(midPt(g(i),g(o(i))))); };

Subdivision in Processing: Split triangles void splitTriangles() { // splits each triangle into 4 for (int i=0; i<3*nt; i=i+3) { V[3*nt+i]=v(i); V[n(3*nt+i)]=w(p(i)); V[p(3*nt+i)]=w(n(i)); V[6*nt+i]=v(n(i)); V[n(6*nt+i)]=w(i); V[p(6*nt+i)]=w(p(i)); V[9*nt+i]=v(p(i)); V[n(9*nt+i)]=w(n(i)); V[p(9*nt+i)]=w(i); V[i]=w(i); V[n(i)]=w(n(i)); V[p(i)]=w(p(i)); }; nt=4*nt;

Subdivision in Processing: Butterfly void keyPressed() { … if (key=='B') { splitEdges(); bulge(); splitTriangles(); computeO(); }; butterfly

Butterfly subdivision & smoothing refine smoothing

Smoothing in Processing: Valences void computeValence() { // caches valence of each vertex // resets the valences to 0 for (int i=0; i<nv; i++) {Nv[i].setTo(0,0,0); valence[i]=0;}; for (int i=0; i<3*nt; i++) {valence[v(i)]++; }; };

Smoothing in Processing: Normals // computes the vertex normals as sums of the normal vectors // of incident tirangles scaled by area/2 void computeVertexNormals() { computeValence(); for (int i=0; i<3*nt; i++) {Nv[v(p(i))].add(g(p(i)).vecTo(g(n(i))));}; for (int i=0; i<nv; i++) {Nv[i].div(valence[i]);}; };

Smoothing in Processing: Tuck // displaces each vertex by a fraction s of its normal void tuck(float s) { for (int i=0; i<nv; i++) {G[i].addScaledVec(s,Nv[i]);}; };

Smoothing in Processing: Smooth void keyPressed() { … // bi-laplace smoothing if (key=='S') { computeVertexNormals(); tuck(0.6); tuck(-0.6); }; smoothing

Modified Butterfly scheme “Interpolating Subdivision for Meshes with Arbitrary Topology”, Zorin, Schröder, Sweldens

Other subdivision schemes Loop subdivision Move mid-edge points 1/4 towards the average of their second neighbors (s) Move the original vertices towards the average of their new neighbors. Loop suggests to use =(5/8-(3/8+(cos(2/k))/4))/k

Adaptive subdivision What if you need to subdivide more in one areas? “A Sub-Atomic Subdivision Approach” by S. Seeger, K. Hormann G. Hausler and G. Greiner

Subdivision of non-triangle meshes Insert mid-edge points. Remove old vertices J. Peters, U. Reif, The Simplest Subdivision Scheme for smoothing polyhedra. 16(4), ToG 97.

What you must remember That regular subdivision multiplies the number of triangles and vertices by 4 That the butterfly subdivision computes edge-midpoints using the averages of first, second, and third neighbors (f, s, t): m(c) = average(f) + (average(s) – average(t))/4 Our high-level approach algorithm for computing the XYZ’ and V’ tables for one level of subdivision. The fact that the butterfly subdivision interpolates the original vertices How to implement an adaptive Butterfly subdivision How to perform a Loop subdivision

Examples of questions for exams How is the midpoint of edge E computed in the butterfly subdivision scheme given a corner c at the opposite tip of one of the two triangles incident upon E? Is the Butterfly subdivision interpolating the original vertices? Justify/prove your answer. How many vertices are introduced by three levels of butterfly subdivision to a zero-genus mesh with V vertices? What is the influence of one of the vertices of the original triangle mesh on the surface resulting from repeatedly applying the regular Butterfly subdivision? (Hint, each one of the original triangles is subdivided into 4, then 16, and so on. Say that repeated subdivision maps a triangle into a curved patch. Obviously, each vertex of the original mesh influences the patches of the triangles incident upon it. Does it influence more?

A Sketch-Based Interface for Detail-Preserving Mesh Editing Andrew Nealen Olga Sorkine Marc Alexa Daniel Cohen-Or

Sketch-Based Interfaces and Modeling (Some) previous work SKETCH [Zeleznik et al. 96] Teddy [Igarashi et al. 99 and 03] Variational implicits [Karpenko et al. 02] Relief [Bourguignon et al. 04] Sketching mesh deformations [Kho and Garland 05] Parametrized objects [Yang et al. 05] Many, many more... and (hopefully) more to come!

Silhouette Sketching Approximate sketching Balance weighting between detail and positional constraints

Silhouette Sketching Approximate sketching Balance weighting between detail and positional constraints

Inspiration and Motivation [Botsch and Kobbelt 04] [Sorkine et al. 04] The (affine) handle metaphor Used in (almost) every editing tool Nice, but can be unintuitive for specific editing tasks Laplacian Mesh Editing Preserve local detail after imposing editing constraints [Sorkine et al. 04] [Zhou et al. 05]

Silhouette Sketching Using silhouettes as handles Detect object space silhouette Project to screen space and parametrize [0,1] Parametrize sketch [0,1] Find correspondences

Silhouette Sketching Using silhouettes as handles Detect object space silhouette Project to screen space and parametrize [0,1] Parametrize sketch [0,1] Find correspondences Use as positional constraints while retaining depth value

Silhouette Sketching What is a good silhouette? viewer